Properties

Label 17.1.861...144.1
Degree $17$
Signature $[1, 8]$
Discriminant $8.615\times 10^{26}$
Root discriminant \(38.41\)
Ramified primes see page
Class number $15$ (GRH)
Class group $[15]$ (GRH)
Galois group $\PSL(2,16)$ (as 17T6)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^17 - x^16 - 4*x^15 + 2*x^14 + 54*x^13 - 6*x^12 - 36*x^11 + 16*x^10 + 714*x^9 + 1238*x^8 + 484*x^7 - 764*x^6 - 1084*x^5 + 520*x^4 + 668*x^3 - 776*x^2 + 382*x - 74)
 
gp: K = bnfinit(x^17 - x^16 - 4*x^15 + 2*x^14 + 54*x^13 - 6*x^12 - 36*x^11 + 16*x^10 + 714*x^9 + 1238*x^8 + 484*x^7 - 764*x^6 - 1084*x^5 + 520*x^4 + 668*x^3 - 776*x^2 + 382*x - 74, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-74, 382, -776, 668, 520, -1084, -764, 484, 1238, 714, 16, -36, -6, 54, 2, -4, -1, 1]);
 

\( x^{17} - x^{16} - 4 x^{15} + 2 x^{14} + 54 x^{13} - 6 x^{12} - 36 x^{11} + 16 x^{10} + 714 x^{9} + 1238 x^{8} + 484 x^{7} - 764 x^{6} - 1084 x^{5} + 520 x^{4} + 668 x^{3} - 776 x^{2} + \cdots - 74 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 

Invariants

Degree:  $17$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[1, 8]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:   \(861526607800060221948166144\) \(\medspace = 2^{30}\cdot 173^{8}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  \(38.41\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:   \(2\), \(173\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$\card{ \Aut(K/\Q) }$:  $1$
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{51\!\cdots\!19}a^{16}-\frac{29\!\cdots\!61}{51\!\cdots\!19}a^{15}-\frac{14\!\cdots\!39}{51\!\cdots\!19}a^{14}+\frac{87\!\cdots\!85}{51\!\cdots\!19}a^{13}+\frac{90\!\cdots\!97}{51\!\cdots\!19}a^{12}+\frac{21\!\cdots\!08}{51\!\cdots\!19}a^{11}+\frac{72\!\cdots\!54}{51\!\cdots\!19}a^{10}+\frac{31\!\cdots\!96}{51\!\cdots\!19}a^{9}+\frac{25\!\cdots\!45}{51\!\cdots\!19}a^{8}-\frac{49\!\cdots\!91}{51\!\cdots\!19}a^{7}-\frac{27\!\cdots\!52}{51\!\cdots\!19}a^{6}+\frac{22\!\cdots\!06}{51\!\cdots\!19}a^{5}-\frac{63\!\cdots\!14}{51\!\cdots\!19}a^{4}+\frac{14\!\cdots\!17}{51\!\cdots\!19}a^{3}+\frac{30\!\cdots\!20}{51\!\cdots\!19}a^{2}+\frac{22\!\cdots\!38}{51\!\cdots\!19}a+\frac{10\!\cdots\!37}{51\!\cdots\!19}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

$C_{15}$, which has order $15$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $8$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:   $\frac{44\!\cdots\!32}{51\!\cdots\!19}a^{16}-\frac{27\!\cdots\!92}{51\!\cdots\!19}a^{15}-\frac{19\!\cdots\!79}{51\!\cdots\!19}a^{14}+\frac{84\!\cdots\!50}{51\!\cdots\!19}a^{13}+\frac{24\!\cdots\!30}{51\!\cdots\!19}a^{12}+\frac{67\!\cdots\!12}{51\!\cdots\!19}a^{11}-\frac{14\!\cdots\!79}{51\!\cdots\!19}a^{10}-\frac{36\!\cdots\!62}{51\!\cdots\!19}a^{9}+\frac{32\!\cdots\!07}{51\!\cdots\!19}a^{8}+\frac{67\!\cdots\!11}{51\!\cdots\!19}a^{7}+\frac{45\!\cdots\!38}{51\!\cdots\!19}a^{6}-\frac{26\!\cdots\!08}{51\!\cdots\!19}a^{5}-\frac{70\!\cdots\!44}{51\!\cdots\!19}a^{4}-\frac{96\!\cdots\!88}{51\!\cdots\!19}a^{3}+\frac{28\!\cdots\!28}{51\!\cdots\!19}a^{2}-\frac{17\!\cdots\!58}{51\!\cdots\!19}a+\frac{75\!\cdots\!83}{51\!\cdots\!19}$, $\frac{21\!\cdots\!47}{51\!\cdots\!19}a^{16}-\frac{12\!\cdots\!41}{51\!\cdots\!19}a^{15}-\frac{93\!\cdots\!11}{51\!\cdots\!19}a^{14}+\frac{43\!\cdots\!83}{51\!\cdots\!19}a^{13}+\frac{11\!\cdots\!10}{51\!\cdots\!19}a^{12}+\frac{37\!\cdots\!85}{51\!\cdots\!19}a^{11}-\frac{69\!\cdots\!45}{51\!\cdots\!19}a^{10}+\frac{39\!\cdots\!07}{51\!\cdots\!19}a^{9}+\frac{15\!\cdots\!48}{51\!\cdots\!19}a^{8}+\frac{33\!\cdots\!96}{51\!\cdots\!19}a^{7}+\frac{24\!\cdots\!55}{51\!\cdots\!19}a^{6}-\frac{82\!\cdots\!39}{51\!\cdots\!19}a^{5}-\frac{29\!\cdots\!47}{51\!\cdots\!19}a^{4}-\frac{20\!\cdots\!35}{51\!\cdots\!19}a^{3}+\frac{14\!\cdots\!31}{51\!\cdots\!19}a^{2}-\frac{11\!\cdots\!92}{51\!\cdots\!19}a+\frac{34\!\cdots\!07}{51\!\cdots\!19}$, $\frac{68\!\cdots\!28}{51\!\cdots\!19}a^{16}+\frac{16\!\cdots\!86}{51\!\cdots\!19}a^{15}-\frac{28\!\cdots\!39}{51\!\cdots\!19}a^{14}-\frac{21\!\cdots\!35}{51\!\cdots\!19}a^{13}+\frac{35\!\cdots\!80}{51\!\cdots\!19}a^{12}+\frac{40\!\cdots\!41}{51\!\cdots\!19}a^{11}+\frac{11\!\cdots\!07}{51\!\cdots\!19}a^{10}+\frac{96\!\cdots\!91}{51\!\cdots\!19}a^{9}+\frac{49\!\cdots\!40}{51\!\cdots\!19}a^{8}+\frac{14\!\cdots\!69}{51\!\cdots\!19}a^{7}+\frac{19\!\cdots\!93}{51\!\cdots\!19}a^{6}+\frac{12\!\cdots\!64}{51\!\cdots\!19}a^{5}+\frac{14\!\cdots\!20}{51\!\cdots\!19}a^{4}-\frac{27\!\cdots\!16}{51\!\cdots\!19}a^{3}+\frac{36\!\cdots\!57}{51\!\cdots\!19}a^{2}-\frac{12\!\cdots\!34}{51\!\cdots\!19}a+\frac{23\!\cdots\!51}{51\!\cdots\!19}$, $\frac{32\!\cdots\!72}{51\!\cdots\!19}a^{16}-\frac{23\!\cdots\!33}{51\!\cdots\!19}a^{15}-\frac{20\!\cdots\!47}{51\!\cdots\!19}a^{14}+\frac{89\!\cdots\!57}{51\!\cdots\!19}a^{13}+\frac{25\!\cdots\!11}{51\!\cdots\!19}a^{12}-\frac{10\!\cdots\!39}{51\!\cdots\!19}a^{11}-\frac{14\!\cdots\!20}{51\!\cdots\!19}a^{10}-\frac{27\!\cdots\!98}{51\!\cdots\!19}a^{9}+\frac{23\!\cdots\!53}{51\!\cdots\!19}a^{8}-\frac{11\!\cdots\!88}{51\!\cdots\!19}a^{7}-\frac{42\!\cdots\!38}{51\!\cdots\!19}a^{6}-\frac{62\!\cdots\!55}{51\!\cdots\!19}a^{5}-\frac{45\!\cdots\!78}{51\!\cdots\!19}a^{4}-\frac{95\!\cdots\!35}{51\!\cdots\!19}a^{3}-\frac{65\!\cdots\!16}{51\!\cdots\!19}a^{2}-\frac{73\!\cdots\!26}{51\!\cdots\!19}a+\frac{16\!\cdots\!27}{51\!\cdots\!19}$, $\frac{15\!\cdots\!79}{51\!\cdots\!19}a^{16}-\frac{83\!\cdots\!62}{51\!\cdots\!19}a^{15}-\frac{67\!\cdots\!22}{51\!\cdots\!19}a^{14}+\frac{19\!\cdots\!51}{51\!\cdots\!19}a^{13}+\frac{84\!\cdots\!23}{51\!\cdots\!19}a^{12}+\frac{29\!\cdots\!81}{51\!\cdots\!19}a^{11}-\frac{50\!\cdots\!52}{51\!\cdots\!19}a^{10}+\frac{46\!\cdots\!34}{51\!\cdots\!19}a^{9}+\frac{11\!\cdots\!52}{51\!\cdots\!19}a^{8}+\frac{24\!\cdots\!11}{51\!\cdots\!19}a^{7}+\frac{17\!\cdots\!71}{51\!\cdots\!19}a^{6}-\frac{53\!\cdots\!23}{51\!\cdots\!19}a^{5}-\frac{19\!\cdots\!94}{51\!\cdots\!19}a^{4}+\frac{34\!\cdots\!29}{51\!\cdots\!19}a^{3}+\frac{12\!\cdots\!49}{51\!\cdots\!19}a^{2}-\frac{74\!\cdots\!71}{51\!\cdots\!19}a+\frac{20\!\cdots\!81}{51\!\cdots\!19}$, $\frac{12\!\cdots\!34}{51\!\cdots\!19}a^{16}-\frac{55\!\cdots\!88}{51\!\cdots\!19}a^{15}-\frac{55\!\cdots\!10}{51\!\cdots\!19}a^{14}-\frac{49\!\cdots\!28}{51\!\cdots\!19}a^{13}+\frac{69\!\cdots\!96}{51\!\cdots\!19}a^{12}+\frac{31\!\cdots\!77}{51\!\cdots\!19}a^{11}-\frac{31\!\cdots\!48}{51\!\cdots\!19}a^{10}+\frac{35\!\cdots\!03}{51\!\cdots\!19}a^{9}+\frac{91\!\cdots\!12}{51\!\cdots\!19}a^{8}+\frac{21\!\cdots\!91}{51\!\cdots\!19}a^{7}+\frac{17\!\cdots\!64}{51\!\cdots\!19}a^{6}-\frac{46\!\cdots\!04}{51\!\cdots\!19}a^{5}-\frac{14\!\cdots\!99}{51\!\cdots\!19}a^{4}-\frac{14\!\cdots\!03}{51\!\cdots\!19}a^{3}+\frac{80\!\cdots\!99}{51\!\cdots\!19}a^{2}-\frac{59\!\cdots\!78}{51\!\cdots\!19}a+\frac{17\!\cdots\!29}{51\!\cdots\!19}$, $\frac{33\!\cdots\!74}{51\!\cdots\!19}a^{16}-\frac{88\!\cdots\!23}{51\!\cdots\!19}a^{15}-\frac{15\!\cdots\!99}{51\!\cdots\!19}a^{14}-\frac{19\!\cdots\!30}{51\!\cdots\!19}a^{13}+\frac{18\!\cdots\!25}{51\!\cdots\!19}a^{12}+\frac{10\!\cdots\!04}{51\!\cdots\!19}a^{11}-\frac{12\!\cdots\!24}{51\!\cdots\!19}a^{10}+\frac{25\!\cdots\!97}{51\!\cdots\!19}a^{9}+\frac{23\!\cdots\!80}{51\!\cdots\!19}a^{8}+\frac{59\!\cdots\!89}{51\!\cdots\!19}a^{7}+\frac{48\!\cdots\!06}{51\!\cdots\!19}a^{6}-\frac{62\!\cdots\!87}{51\!\cdots\!19}a^{5}-\frac{48\!\cdots\!83}{51\!\cdots\!19}a^{4}-\frac{12\!\cdots\!77}{51\!\cdots\!19}a^{3}+\frac{37\!\cdots\!60}{51\!\cdots\!19}a^{2}-\frac{19\!\cdots\!86}{51\!\cdots\!19}a+\frac{73\!\cdots\!79}{51\!\cdots\!19}$, $\frac{36\!\cdots\!06}{51\!\cdots\!19}a^{16}-\frac{78\!\cdots\!91}{51\!\cdots\!19}a^{15}-\frac{62\!\cdots\!61}{51\!\cdots\!19}a^{14}+\frac{78\!\cdots\!32}{51\!\cdots\!19}a^{13}+\frac{18\!\cdots\!98}{51\!\cdots\!19}a^{12}-\frac{13\!\cdots\!02}{51\!\cdots\!19}a^{11}+\frac{37\!\cdots\!44}{51\!\cdots\!19}a^{10}-\frac{86\!\cdots\!77}{51\!\cdots\!19}a^{9}+\frac{26\!\cdots\!17}{51\!\cdots\!19}a^{8}+\frac{31\!\cdots\!15}{51\!\cdots\!19}a^{7}-\frac{30\!\cdots\!55}{51\!\cdots\!19}a^{6}-\frac{11\!\cdots\!90}{51\!\cdots\!19}a^{5}-\frac{20\!\cdots\!47}{51\!\cdots\!19}a^{4}-\frac{58\!\cdots\!75}{51\!\cdots\!19}a^{3}+\frac{39\!\cdots\!91}{51\!\cdots\!19}a^{2}-\frac{30\!\cdots\!87}{51\!\cdots\!19}a+\frac{22\!\cdots\!57}{51\!\cdots\!19}$ Copy content Toggle raw display (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 781146.759847 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{1}\cdot(2\pi)^{8}\cdot 781146.759847 \cdot 15}{2\sqrt{861526607800060221948166144}}\approx 0.969680124809$ (assuming GRH)

Galois group

$\PSL(2,16)$ (as 17T6):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A non-solvable group of order 4080
The 17 conjugacy class representatives for $\PSL(2,16)$
Character table for $\PSL(2,16)$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R $17$ $17$ $17$ $17$ $15{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ $17$ $17$ $15{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ $15{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ $15{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ ${\href{/padicField/37.5.0.1}{5} }^{3}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ $15{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ $17$ $17$ ${\href{/padicField/53.3.0.1}{3} }^{5}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ ${\href{/padicField/59.2.0.1}{2} }^{8}{,}\,{\href{/padicField/59.1.0.1}{1} }$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 
magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display $\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
Deg $16$$16$$1$$30$
\(173\) Copy content Toggle raw display $\Q_{173}$$x + 2$$1$$1$$0$Trivial$[\ ]$
173.4.2.1$x^{4} + 1557 x^{2} + 748225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
173.4.2.1$x^{4} + 1557 x^{2} + 748225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
173.4.2.1$x^{4} + 1557 x^{2} + 748225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
173.4.2.1$x^{4} + 1557 x^{2} + 748225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
15.861...144.240.a.a$15$ $ 2^{30} \cdot 173^{8}$ 17.1.861526607800060221948166144.1 $\PSL(2,16)$ (as 17T6) $1$ $-1$
15.861...144.240.a.b$15$ $ 2^{30} \cdot 173^{8}$ 17.1.861526607800060221948166144.1 $\PSL(2,16)$ (as 17T6) $1$ $-1$
15.861...144.240.a.c$15$ $ 2^{30} \cdot 173^{8}$ 17.1.861526607800060221948166144.1 $\PSL(2,16)$ (as 17T6) $1$ $-1$
15.861...144.240.a.d$15$ $ 2^{30} \cdot 173^{8}$ 17.1.861526607800060221948166144.1 $\PSL(2,16)$ (as 17T6) $1$ $-1$
15.861...144.240.a.e$15$ $ 2^{30} \cdot 173^{8}$ 17.1.861526607800060221948166144.1 $\PSL(2,16)$ (as 17T6) $1$ $-1$
15.861...144.240.a.f$15$ $ 2^{30} \cdot 173^{8}$ 17.1.861526607800060221948166144.1 $\PSL(2,16)$ (as 17T6) $1$ $-1$
15.861...144.240.a.g$15$ $ 2^{30} \cdot 173^{8}$ 17.1.861526607800060221948166144.1 $\PSL(2,16)$ (as 17T6) $1$ $-1$
15.861...144.240.a.h$15$ $ 2^{30} \cdot 173^{8}$ 17.1.861526607800060221948166144.1 $\PSL(2,16)$ (as 17T6) $1$ $-1$
* 16.861...144.17t6.a.a$16$ $ 2^{30} \cdot 173^{8}$ 17.1.861526607800060221948166144.1 $\PSL(2,16)$ (as 17T6) $1$ $0$
17.861...144.51.a.a$17$ $ 2^{30} \cdot 173^{8}$ 17.1.861526607800060221948166144.1 $\PSL(2,16)$ (as 17T6) $1$ $1$
17.861...144.68.a.a$17$ $ 2^{30} \cdot 173^{8}$ 17.1.861526607800060221948166144.1 $\PSL(2,16)$ (as 17T6) $1$ $1$
17.861...144.68.a.b$17$ $ 2^{30} \cdot 173^{8}$ 17.1.861526607800060221948166144.1 $\PSL(2,16)$ (as 17T6) $1$ $1$
17.861...144.120.a.a$17$ $ 2^{30} \cdot 173^{8}$ 17.1.861526607800060221948166144.1 $\PSL(2,16)$ (as 17T6) $1$ $1$
17.861...144.120.a.b$17$ $ 2^{30} \cdot 173^{8}$ 17.1.861526607800060221948166144.1 $\PSL(2,16)$ (as 17T6) $1$ $1$
17.861...144.120.a.c$17$ $ 2^{30} \cdot 173^{8}$ 17.1.861526607800060221948166144.1 $\PSL(2,16)$ (as 17T6) $1$ $1$
17.861...144.120.a.d$17$ $ 2^{30} \cdot 173^{8}$ 17.1.861526607800060221948166144.1 $\PSL(2,16)$ (as 17T6) $1$ $1$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.